Embedded newform 5.35.c.a.2.1

• Label: 5.35.c.a.2.1
• Level: $5$
• Weight: $35$
• Character: 5.2
• Analytic conductor: $36.613$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 35 \)
Character orbit:	\([\chi]\)	\(=\)	5.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(36.6128270213\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2.1
Character	\(\chi\)	\(=\)	5.2
Dual form			5.35.c.a.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-169510. - 169510. i) q^{2} +(-198644. + 198644. i) q^{3} +4.02874e10i q^{4} +(-3.95922e11 + 6.52168e11i) q^{5} +6.73444e10 q^{6} +(1.10969e14 + 1.10969e14i) q^{7} +(3.91696e15 - 3.91696e15i) q^{8} +1.66771e16i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	32 * q + 131070 * q^2 - 78988260 * q^3 - 730721770860 * q^5 - 44043049959816 * q^6 + 5337954235100 * q^7 - 4536119823236220 * q^8

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−169510.	−	169510.i	−1.29326	−	1.29326i	−0.932759	−	0.360499i	\(-0.882606\pi\)
							−0.360499	−	0.932759i	\(-0.617394\pi\)
\(3\)	−198644.	+	198644.i	−0.00153821	+	0.00153821i	−0.707875	−	0.706337i	\(-0.750346\pi\)
							0.706337	+	0.707875i	\(0.250346\pi\)
\(5\)	−3.95922e11	+	6.52168e11i	−0.518942	+	0.854809i
							\(7\)	1.10969e14	+	1.10969e14i	0.477020	+	0.477020i	0.904177	−	0.427157i	\(-0.140485\pi\)
−0.427157	+	0.904177i	\(0.640485\pi\)
\(11\)	8.70815e17			1.72286			0.861431	−	0.507875i	\(-0.169569\pi\)
							0.861431	+	0.507875i	\(0.169569\pi\)
\(13\)	−7.62676e18	+	7.62676e18i	−0.881664	+	0.881664i	−0.993704	−	0.112040i	\(-0.964261\pi\)
							0.112040	+	0.993704i	\(0.464261\pi\)
\(17\)	8.05760e19	+	8.05760e19i	0.0974034	+	0.0974034i	0.754129	−	0.656726i	\(-0.228059\pi\)
							−0.656726	+	0.754129i	\(0.728059\pi\)
\(19\)			7.48550e20i			0.136587i	0.997665	+	0.0682936i	\(0.0217554\pi\)
							−0.997665	+	0.0682936i	\(0.978245\pi\)
\(23\)	−7.83761e22	+	7.83761e22i	−0.555661	+	0.555661i	−0.928069	−	0.372408i	\(-0.878532\pi\)
							0.372408	+	0.928069i	\(0.378532\pi\)
\(29\)			9.02227e24i			1.24323i	0.783325	+	0.621613i	\(0.213522\pi\)
							−0.783325	+	0.621613i	\(0.786478\pi\)
\(31\)	3.35683e25			1.48861			0.744304	−	0.667841i	\(-0.232781\pi\)
							0.744304	+	0.667841i	\(0.232781\pi\)
\(37\)	−1.09301e26	−	1.09301e26i	−0.239439	−	0.239439i	0.577179	−	0.816618i	\(-0.304154\pi\)
							−0.816618	+	0.577179i	\(0.804154\pi\)
\(41\)	3.70169e27			1.41604			0.708019	−	0.706194i	\(-0.249589\pi\)
							0.708019	+	0.706194i	\(0.249589\pi\)
\(43\)	−2.65186e27	+	2.65186e27i	−0.451426	+	0.451426i	−0.895828	−	0.444402i	\(-0.853416\pi\)
							0.444402	+	0.895828i	\(0.353416\pi\)
\(47\)	−1.63651e27	−	1.63651e27i	−0.0614122	−	0.0614122i	0.675734	−	0.737146i	\(-0.263827\pi\)
							−0.737146	+	0.675734i	\(0.763827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.35.c.a.2.1	✓	32
5.2	odd	4		25.35.c.b.18.16		32
5.3	odd	4	inner	5.35.c.a.3.1	yes	32
5.4	even	2		25.35.c.b.7.16		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.35.c.a.2.1	✓	32	1.1	even	1	trivial
5.35.c.a.3.1	yes	32	5.3	odd	4	inner
25.35.c.b.7.16		32	5.4	even	2
25.35.c.b.18.16		32	5.2	odd	4